142 research outputs found
Non-Abelian Analogs of Lattice Rounding
Lattice rounding in Euclidean space can be viewed as finding the nearest
point in the orbit of an action by a discrete group, relative to the norm
inherited from the ambient space. Using this point of view, we initiate the
study of non-abelian analogs of lattice rounding involving matrix groups. In
one direction, we give an algorithm for solving a normed word problem when the
inputs are random products over a basis set, and give theoretical justification
for its success. In another direction, we prove a general inapproximability
result which essentially rules out strong approximation algorithms (i.e., whose
approximation factors depend only on dimension) analogous to LLL in the general
case.Comment: 30 page
Non-Abelian Analogs of Lattice Rounding
Lattice rounding in Euclidean space can be viewed as finding the nearest point in the orbit of an action by a discrete group, relative to the norm inherited from the ambient space. Using this point of view, we initiate the study of non-abelian analogs of lattice rounding involving matrix groups. In one direction, we give an algorithm for solving a normed word problem when the inputs are random products over a basis set, and give theoretical justification for its success. In another direction, we prove a general inapproximability result which essentially rules out strong approximation algorithms (i.e., whose approximation factors depend only on dimension) analogous to LLL in the general case
The Phase Diagram of Compact QED Coupled to a Four-Fermi Interaction
Compact lattice Quantum Electrodynamics (QED) with four species of fermions
is simulated with massless quarks by using the QED scheme of adding a
four-fermi interaction to the action. Simulations directly in the chiral limit
of massless quarks are done with high statistics on , and lattices,
and the phase diagram, parameterized by the gauge and the four-fermi couplings,
is mapped out. The line of monopole condensation transitions is separate from
the line of chiral symmetry restoration. The simulation results indicate that
the monopole condensation transition is first order while the chiral transition
is second order. The challenges in determining the Universality class of the
chiral transition are discussed. If the scaling region for the chiral
transition is sufficiently wide, the simulations predict critical
indices far from mean field values. We discuss a speculative scenario in which
anti-screening provided by double-helix strands of monopole and anti-monopole
loops are the agent that balances the screening of fermion anti-fermion pairs
to produce an ultra-violet fixed point in the electric coupling.Comment: 29 pages, 8 figures and 2 table
The Gromov width of 4-dimensional tori
We show that every 4-dimensional torus with a linear symplectic form can be
fully filled by one symplectic ball. If such a torus is not symplectomorphic to
a product of 2-dimensional tori with equal sized factors, then it can also be
fully filled by any finite collection of balls provided only that their total
volume is less than that of the 4-torus with its given linear symplectic form.Comment: improved exposition, proof of Proposition 3.9 clarified, discussion
of ellipsoid embeddings remove
Non-Abelian chiral spin liquid in a quantum antiferromagnet revealed by an iPEPS study
Abelian and non-Abelian topological phases exhibiting protected chiral edge
modes are ubiquitous in the realm of the Fractional Quantum Hall (FQH) effect.
Here, we investigate a spin-1 Hamiltonian on the square lattice which could,
potentially, host the spin liquid analog of the (bosonic) non-Abelian
Moore-Read FQH state, as suggested by Exact Diagonalisation of small clusters.
Using families of fully SU(2)-spin symmetric and translationally invariant
chiral Projected Entangled Pair States (PEPS), variational energy optimization
is performed using infinite-PEPS methods, providing good agreement with Density
Matrix Renormalisation Group (DMRG) results. A careful analysis of the bulk
spin-spin and dimer-dimer correlation functions in the optimized spin liquid
suggests that they exhibit long-range "gossamer tails". We argue these tails
are finite- artifacts of the chiral PEPS, which become irrelevant when the
PEPS bond dimension is increased. From the investigation of the
entanglement spectrum, we observe sharply defined chiral edge modes following
the prediction of the SU(2) Wess-Zumino-Witten theory and exhibiting a
conformal field theory (CFT) central charge , as expected for a
Moore-Read chiral spin liquid. We conclude that the PEPS formalism offers an
unbiased and efficient method to investigate non-Abelian chiral spin liquids in
quantum antiferromagnets.Comment: 15 pages, 16 figures - references added - small changes in title and
abstrac
Analysis of sum-of-squares relaxations for the quantum rotor model
The noncommutative sum-of-squares (ncSoS) hierarchy was introduced by
Navascu\'{e}s-Pironio-Ac\'{i}n as a sequence of semidefinite programming
relaxations for approximating values of noncommutative polynomial optimization
problems, which were originally intended to generalize quantum values of
nonlocal games. Recent work has started to analyze the hierarchy for
approximating ground energies of local Hamiltonians, initially through rounding
algorithms which output product states for degree-2 ncSoS applied to Quantum
Max-Cut. Some rounding methods are known which output entangled states, but
they use degree-4 ncSoS. Based on this, Hwang-Neeman-Parekh-Thompson-Wright
conjectured that degree-2 ncSoS cannot beat product state approximations for
Quantum Max-Cut and gave a partial proof relying on a conjectural
generalization of Borrell's inequality. In this work we consider a family of
Hamiltonians (called the quantum rotor model in condensed matter literature or
lattice vector model in quantum field theory) with infinite-dimensional
local Hilbert space , and show that a degree-2 ncSoS
relaxation approximates the ground state energy better than any product state.Comment: 28 pages, submitted to QIP 202
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